Homoclinic solutions for a class of autonomous second order Hamiltonian systems with a superquadratic potential

Abstract

We will prove the existence of a nontrivial homoclinic solution for an autonomous second order Hamiltonian system $\ddot{q}+\nabla{V}(q)=0$, where $q\in\mathbb{R}^n$, a potential $V\colon\mathbb{R}^n\to\mathbb{R}$ is of the form $V(q)=-K(q)+W(q)$, $K$ and $W$ are $C^{1}$-maps, $K$ satisfies the pinching condition, $W$ grows at a superquadratic rate, as $|q|\to\infty$ and $W(q)=o(|q|^2)$, as $|q|\to 0$. A homoclinic solution will be obtained as a weak limit in the Sobolev space $W^{1,2}(\mathbb{R},\mathbb{R}^n)$ of a sequence of almost critical points of the corresponding action functional. Before passing to a weak limit with a sequence of almost critical points each element of this sequence has to be appropriately shifted.